Integrand size = 45, antiderivative size = 19 \[ \int \frac {16 x-8 x^3+\left (-24 x^2-12 x^3\right ) \log (x)+\left (-24 x^2-12 x^3\right ) \log (2+x)}{2+x} \, dx=4 x^2 (1-x \log (x)-x \log (2+x)) \]
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Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.178, Rules used = {6820, 12, 6874, 14, 786, 2341, 2442, 45} \[ \int \frac {16 x-8 x^3+\left (-24 x^2-12 x^3\right ) \log (x)+\left (-24 x^2-12 x^3\right ) \log (2+x)}{2+x} \, dx=-4 x^3 \log (x)-4 x^3 \log (x+2)+4 x^2 \]
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Rule 12
Rule 14
Rule 45
Rule 786
Rule 2341
Rule 2442
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {4 x \left (-2 \left (-2+x^2\right )-3 x (2+x) \log (x)-3 x (2+x) \log (2+x)\right )}{2+x} \, dx \\ & = 4 \int \frac {x \left (-2 \left (-2+x^2\right )-3 x (2+x) \log (x)-3 x (2+x) \log (2+x)\right )}{2+x} \, dx \\ & = 4 \int \left (-\frac {x \left (-4+2 x^2+6 x \log (x)+3 x^2 \log (x)\right )}{2+x}-3 x^2 \log (2+x)\right ) \, dx \\ & = -\left (4 \int \frac {x \left (-4+2 x^2+6 x \log (x)+3 x^2 \log (x)\right )}{2+x} \, dx\right )-12 \int x^2 \log (2+x) \, dx \\ & = -4 x^3 \log (2+x)+4 \int \frac {x^3}{2+x} \, dx-4 \int x \left (\frac {2 \left (-2+x^2\right )}{2+x}+3 x \log (x)\right ) \, dx \\ & = -4 x^3 \log (2+x)+4 \int \left (4-2 x+x^2-\frac {8}{2+x}\right ) \, dx-4 \int \left (\frac {2 x \left (-2+x^2\right )}{2+x}+3 x^2 \log (x)\right ) \, dx \\ & = 16 x-4 x^2+\frac {4 x^3}{3}-32 \log (2+x)-4 x^3 \log (2+x)-8 \int \frac {x \left (-2+x^2\right )}{2+x} \, dx-12 \int x^2 \log (x) \, dx \\ & = 16 x-4 x^2+\frac {8 x^3}{3}-4 x^3 \log (x)-32 \log (2+x)-4 x^3 \log (2+x)-8 \int \left (2-2 x+x^2-\frac {4}{2+x}\right ) \, dx \\ & = 4 x^2-4 x^3 \log (x)-4 x^3 \log (2+x) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {16 x-8 x^3+\left (-24 x^2-12 x^3\right ) \log (x)+\left (-24 x^2-12 x^3\right ) \log (2+x)}{2+x} \, dx=-4 \left (-x^2+x^3 \log (x)+x^3 \log (2+x)\right ) \]
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Time = 0.87 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21
method | result | size |
risch | \(-4 x^{3} \ln \left (x \right )-4 x^{3} \ln \left (2+x \right )+4 x^{2}\) | \(23\) |
parallelrisch | \(-4 x^{3} \ln \left (x \right )-4 x^{3} \ln \left (2+x \right )+4 x^{2}\) | \(23\) |
default | \(-4 x^{3} \ln \left (x \right )-4 \left (2+x \right )^{3} \ln \left (2+x \right )+4 x^{2}+\frac {176}{3}+24 \left (2+x \right )^{2} \ln \left (2+x \right )-48 \left (2+x \right ) \ln \left (2+x \right )+32 \ln \left (2+x \right )\) | \(52\) |
parts | \(-4 x^{3} \ln \left (x \right )-4 \left (2+x \right )^{3} \ln \left (2+x \right )+4 x^{2}+\frac {176}{3}+24 \left (2+x \right )^{2} \ln \left (2+x \right )-48 \left (2+x \right ) \ln \left (2+x \right )+32 \ln \left (2+x \right )\) | \(52\) |
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Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {16 x-8 x^3+\left (-24 x^2-12 x^3\right ) \log (x)+\left (-24 x^2-12 x^3\right ) \log (2+x)}{2+x} \, dx=-4 \, x^{3} \log \left (x + 2\right ) - 4 \, x^{3} \log \left (x\right ) + 4 \, x^{2} \]
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Time = 0.21 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.68 \[ \int \frac {16 x-8 x^3+\left (-24 x^2-12 x^3\right ) \log (x)+\left (-24 x^2-12 x^3\right ) \log (2+x)}{2+x} \, dx=- 4 x^{3} \log {\left (x \right )} + 4 x^{2} + \left (- 4 x^{3} - 8\right ) \log {\left (x + 2 \right )} + 8 \log {\left (x + 2 \right )} \]
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Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.58 \[ \int \frac {16 x-8 x^3+\left (-24 x^2-12 x^3\right ) \log (x)+\left (-24 x^2-12 x^3\right ) \log (2+x)}{2+x} \, dx=-4 \, x^{3} \log \left (x\right ) + 4 \, x^{2} - 4 \, {\left (x^{3} + 8\right )} \log \left (x + 2\right ) + 32 \, \log \left (x + 2\right ) \]
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Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {16 x-8 x^3+\left (-24 x^2-12 x^3\right ) \log (x)+\left (-24 x^2-12 x^3\right ) \log (2+x)}{2+x} \, dx=-4 \, x^{3} \log \left (x + 2\right ) - 4 \, x^{3} \log \left (x\right ) + 4 \, x^{2} \]
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Time = 13.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {16 x-8 x^3+\left (-24 x^2-12 x^3\right ) \log (x)+\left (-24 x^2-12 x^3\right ) \log (2+x)}{2+x} \, dx=4\,x^2-4\,x^3\,\ln \left (x+2\right )-4\,x^3\,\ln \left (x\right ) \]
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